To determine the coefficient of [tex]\( x^2 \)[/tex] in the expression [tex]\( (3x + x^3) \left( x + \frac{1}{x} \right) \)[/tex], we need to expand the product and then identify the term that contains [tex]\( x^2 \)[/tex]. Let's break this down step-by-step.
1. Start with the initial expressions:
[tex]\[
(3x + x^3) \quad \text{and} \quad \left( x + \frac{1}{x} \right).
\][/tex]
2. Distribute the terms in the first expression across the terms in the second expression:
[tex]\[
(3x + x^3) \left( x + \frac{1}{x} \right) = 3x \cdot x + 3x \cdot \frac{1}{x} + x^3 \cdot x + x^3 \cdot \frac{1}{x}.
\][/tex]
3. Multiply the terms:
[tex]\[
3x \cdot x = 3x^2,
\][/tex]
[tex]\[
3x \cdot \frac{1}{x} = 3,
\][/tex]
[tex]\[
x^3 \cdot x = x^4,
\][/tex]
[tex]\[
x^3 \cdot \frac{1}{x} = x^2.
\][/tex]
4. Combine all the terms to get the expanded expression:
[tex]\[
3x^2 + 3 + x^4 + x^2.
\][/tex]
5. Group like terms:
[tex]\[
3x^2 + x^2 + 3 + x^4 = 4x^2 + x^4 + 3.
\][/tex]
6. Identify the coefficient of [tex]\( x^2 \)[/tex]:
[tex]\[
\text{The coefficient of } x^2 \text{ is } 4.
\][/tex]
Therefore, the coefficient of [tex]\( x^2 \)[/tex] in [tex]\( (3x + x^3) \left( x + \frac{1}{x} \right) \)[/tex] is [tex]\(\boxed{4}\)[/tex].
